url: Special Issue ijpam.eu A Complete ...acadpubl.eu/jsi/2017-114-5/articles/2/31.pdf · used to generalize the behavior of genetic approach [3]. In most of the evolutionary and

A Complete Framework ForMulti-Constrained 3D Bin Packing

Optimization Using Firefly Algorithm

S.K.Rajesh Kanna1 and K.C.Udaiyakumar2

1Professor, Rajalakshmi Engineering College,Chennai, India.

2Vice Principal, SRM University,Ramapuram Campus, Chennai, India.

E-mail:[email protected]

March 6, 2017

Abstract

In this paper, Firefly Algorithm is used to solve 3D pack-ing of arbitrary sized heterogeneous bins into a container ofstandard size, by considering packing constraints namelyplacement constraint, overlapping constraint, stability con-straint, weight constraint, load bearing constraint and ori-entation constraint. The main objective of this research isto optimally pack four different shapes of bins namely cube,rectangular prism, cylinder and sphere of varying sizes intoa container of standard dimension by meeting the packingconstraints. Firefly algorithm is one of the several natureinspired algorithms which mimics the flashing behavior ofthe fireflies. The flashing behavior is to attract brighterfireflies in the group for mating. The group of fireflies andits brightness are mapped to the set of available bins andthe volume occupied by a set of bins respectively. The per-formance had been evaluated by testing on three classes ofbenchmark data set from OR Library. The results haveshown that the firefly algorithm performance is compara-tively better for most of the problem cases.

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International Journal of Pure and Applied MathematicsVolume 114 No. 6 2017, 267 - 282ISSN: 1311-8080 (printed version); ISSN: 1314-3395 (on-line version)url: http://www.ijpam.euSpecial Issue ijpam.eu

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Key Words and Phrases: Firefly Algorithm; Binpacking problem; Multi-constraint; Optimization.

1 Introduction

Due to globalization and revolution in information technology, firmsare facing a fierce competition and there are many situations inwhich decisions need to be made without knowing their completeimplications. So heuristics are being developed separately for spe-cific domains through trial and error process to make decisions. Onesuch category is the globalized online marketing scenario, there aregiven an unlimited supply of variety of products of different cate-gories ordered by diverse costumers all over the world from differentmanufacturers. All the ordered products of volume ’V’, shape ’S’and weight ’W’ should be delivered to their customers in the short-est delivery time to sustain in the globalized market. Generally, thedelivery processes are carryout by the logistic sectors. These logis-tic firms packed those products from the supplier into cartoon orwooden bins and those bins are packed into containers of standardsize to transport to the required customer location through air /road / ship cargos.

The logistic cost Ci associated with the packing are standardfreight cost for using the container and variable cost depends uponvolume and weight of the product along with the accessories likespacers, pallets, etc. This packing cost also added with the productcost without adding any value to the product. So this non-valueadded cost should be minimized by optimally packing bins into thecontainer and by maximally utilizing the available container vol-ume. Every bin having its own packing constraints like placementconstraint, overlapping constraint, stability constraint, weight con-straint, load bearing constraint, orientation constraint, etc. Theseconstraints should be considered while packing the bins, to get afeasible packing solution. Thus the bin packing problem with con-straint satisfaction became more complex and categorized underNP hard problems.

The researchers are also made greater efforts to solve the NPhard problems using techniques based on swarm intelligence, na-ture inspired algorithms, artificial intelligence, heuristics, etc. The

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swarm intelligence is the interaction between agents from the samepopulation and also with the surrounding environment rather thanrandom search [2]. Nature inspired algorithms mimic the behav-ior of the nature in identifying the optimal solution like ant colonyalgorithm, which mimic the behavior of ants in searching the foodpath [13], Firefly algorithm, which mimic the behavior of firefly inidentifying its mating pair [23], etc. Firefly luciferase is an exam-ple of bioluminescent which can converts the chemical energy intolight rays and brightness of the light intensity attracts the nearbyfireflies [1]. Also recent research proved that the firefly algorithmproducing result at par [14, 15, 17, 18, 19, 22].

So in this research, Firefly algorithm had been used to iden-tify the optimal bin packing sequence for a container by satisfyingthe packing constraints such as placement constraint, overlappingconstraint, stability constraint, weight constraint, load bearing con-straint and orientation constraint. The structure of the paper is asfollows: Section 2 gives a brief review into the literature of exist-ing research on bin packing and firefly algorithm. Section 3 givesthe details of bin packing problem and the constraints involved inpacking. Section 4 explores the firefly algorithm and its parame-ters. The experimental methodology is explained in Section 5 alongwith main results in Section 6. Section 7 concludes the results andtheir implications.

2 Literature Survey

This section covers the research work carried out in the area ofbin packing problem and relevant issues, followed by the fireflyalgorithm implementations for solving optimization problems.

2.1 Bin Packing

Bin packing problem or container loading problem is a well knownNP-hard combinatorial optimization problem which requires pack-ing of number of bins with given sizes into a minimal number offixed capacity containers for transportation [4]. Study on bin pack-ing started in the early 1970s [8, 9] with single type of bins withthe size and cost of a bin were assumed to unity. In the early1980s, many approximation schemes were developed and most of

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them are partitioned the bins from smaller to larger items [10]. Inmost cases, approximate solutions were found for the large binsand the smaller bins packed in the remaining empty space greedily.But the bin packing with constraints are harder than the basic binpacking, because smaller bins may have packing constraints. Foronline bin packing, immediate decisions have to be taken for pack-ing of arriving bins of arbitrary sizes into partially filled standardcapacity containers [6, 7]. Bin packing were solved using the ge-netic approach and found that performs notably better than theheuristics such as first and best fit [5, 12, 15]. The cutting stockproblem and the bin packing problems are resembles the same andmany research also carried out in cutting stock area. Best fit andparallel beam strategy had been hybridized to identify the 2D ori-entation of the cutting stock [11]. Applying a parameter tuningapproach might not match the performance of genetic algorithmin the overall and on the other hand, k-means clustering can beused to generalize the behavior of genetic approach [3]. In most ofthe evolutionary and heuristic algorithms, the major problem is thesetting of prompt value for the algorithm control parameters [26].Some of the researchers use the segment tree for 2D and quad treefor placing the 3D bins into the container and thereby reducing thecomputation time in identifying the best placement position [20].So in this research, bin packing problem had solved by consideringsingle container with four different shapes of arbitrary sized binswith packing constraints using firefly algorithm.

2.2 Firefly Algorithm

In this paper, potentially conflicting constraints and multi-objectivefunctions have been considered and then investigated various pa-rameter settings for the firefly algorithm to result feasible optimalbin packing sequence. Firefly algorithm is developed and show su-periority over some traditional algorithms [18] and since its ap-pearance, the firefly algorithm has shown its promising effective-ness for various optimization problems [14]. The Firefly Algorithmwas developed by Xin She Yang [24, 25] in late 2007 and 2008 inCambridge University. Wang et al [19], used the modified fireflyalgorithm to solve the uninhabited combat air vehicle UCAV pathplanning problem. Sulaiman et al [17], presented a modified Fire-

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fly Algorithm for solving economic dispatch problems as one of themost challenging problems of power system. Firefly algorithm hadalso used to solve the traffic grooming problem and its quality hadcompared with the other swam intelligence algorithms [2]. Fromthe literature survey, it became clear that the firefly algorithm pro-duces better results in many applications and it has been used inthis research to identify the optimal solution for bin packing prob-lem.

3 Bin packing problem

Generally, the products to be transported are packed into the boxesor cartoons called bins. The shape, size and orientation of the binsdepend on the nature of the product. These bins are packed intothe container of standard fixed size by the logistic industries. In or-der to give a formal definition of the problem, let V is the containervolume and the objective is to maximally utilize the available vol-ume without empty spaces inside the container Vmax i.e. Vmax ≈ V.N is the total number of bins available for packing and C is theconstraints in packing the bins. u be any upper bound for the po-tential bins to be packed and are numbered as 1, ..., u. Because inthe single container loading problem, all N bins cannot be packedinto a container. The objective function is given in the equation 1.

Max.f(x) = V (B) − C(B) (1)

V (B) =n∑

i=1

(Li × Bi × Hi)/(V (C)) (2)

C(B) = P (X) + O(X) + S(X) + Wt(X) + R(X) + LB(X)(3)

Whereas, f(x) is the maximization fitness function which calcu-lates the volume occupied inside the container by the packed bins,C(B) is the penalty function,

C(B) = 1.00, if bins violates O(X) and LB(X)

= 0.50, if bins violates P (X) and LB(X)

= 0.25, if a bin violates S(X),Wt(X), R(X)

= 0, Otherwise.

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Whereas, P (X) is the Placement (Boundary crossing) constraintwhich checks the placement of the bins within the container bound-ary and also the placement of the bins based on the unloadingsequence. O(X), S(X),Wt(X), R(X) and LB(X) represents theOverlapping constraint, Stability constraint, Weight constraint, Ori-entation constraint and Load bearing constraint respectively. Over-lapping constraint checks the overlapping among the bins, bins withcontainer, bins with spacers/pallets and other overlapping. Stabil-ity constraint checks the stability of the bins packed over the otherto avoid the collision and roll over during cargo transport. Weightconstraint checks the weight carrying capacity of the each bin ineach layer along with the fragility. Orientation constraint is theuser defined constraint for the product orientation and should befollowed while packing. Load bearing constraint checks with themaximum pay load of the container and freight rate limitation.

A problem instance considered in this research is a sequence of1000 integer values, each representing a bin drawn in order as storedin the database. The bins are sorted based on the weight valuesand fragility. The bins are then numbered in order. There are twoways of setting the number of instances in a sequence. A commonusage is to create a sequence of fixed instance and pack the bins inorder inside the container, till the container packed fully and theremaining bins in the sequence became unpacked. The second wayis generating dynamic number of instance and recuressily packingthe bins. In this reseaarch, fixed number of instance was createdthrough sensitivity analysis. In this research, the bins to be packedinside a container are defined using a matrix, which will have sim-ilar bins in a column. F represents a fitness matrix of search spacefor a given problem and Fr,s is a fitness integer score at the rth

row and sth column of the matrix, where Fr,s ∈ [0, 1]. Fr,s gives thefeasible bin packing score by considering the packing constraints.If Fr,s = 1, represents the maximum utilization of container vol-ume, else if Fr,s = 0 or negative, represents the constraint violationand unfeasible solution. The remaining fitness score represents thepercentage utilization of the container volume. The scores in eachcolumn is independent from that of other columns and each columncan be quite different than that of remaining columns. Also thescore depends on the score of the packed bins and its constraints.Thus the problem became more complex. So for identifying the

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optimal bin packing sequence, firefly algorithm have been used inthis research.

4 Firefly Algorithm

Firefly Algorithm (FFA) is a meta-heuristic biologically inspiredpopulation-based optimization algorithm developed by Xin-She Yangin 2007 [21, 22, 23]. Bioluminescence is a biochemical process whichproduces flashing patterns and the firefly with lower brightness willbe attracted towards the brighter intensity firefly [21]. The follow-ing assumptions are made to map the flashing characteristics offireflies with the practical bin packing problems [22].

a) All the fireflies are unisex, so each firefly can be attracted to-wards all other fireflies.

b) The attractiveness is proportional to their brightness of the lightintensity. Thus, firefly with lesser brightness will be attractedtowards the higher brightness. Brightness depends on the dis-tance between the fireflies and the surrounding environment.

c) The brightness of the light intensity is determined by the objec-tive function.

A firefly’s light intensity or brightness can be calculated from theEquation 4.

I(r) = I0e−Y r (4)

Whereas, I0 is the current light intensity of the firefly, Y is theabsorption coefficient, and r is the distance between ith and jth

fireflies.From the Equation 4, light intensity is exponentially propor-

tional to the distance and the absorption coefficient. i.e. lightintensity decreases with the increase in the distance and absorp-tion. As the result, the attractiveness is changing with the degreeof absorption [16].

The attractiveness (β) is relatively proportional to light inten-sity (I) being observed by the other fireflies, the attractiveness iscalculated using the Equation 5.

β = β0e−Y r2 (5)

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Whereas, β0 is the attractiveness at distance rij = 0.The distance between ith and jth fireflies at Xi and Xj positions

are expressed in the Euclidean form in the Equation 6[24],

rij =∥ Xi − Xj ∥={∑

(Xik − Xjk)2}(1/2)

, k = {1, 2, ..., d} (6)

Whereas, rij denotes distance between ith and jth fireflies, Xi,k isthe kth of Xi, which are the problem dimensions.

Movement of firefly ith towards jth firefly is stated in the Equa-tion 7.

Xi = Xi + β0e−yr2(Xi − Xj) + aϵi (7)

Whereas, Xi is the existing position of ith firefly, a is a random-ization parameter [0,1], and ϵ is a random number [0,1], Gaus-sian distribution which decides the convergence rate. The proposedmethodology is explained in the following section.

5 Experimental implementation

In this research, solving the bin packing using FFA had been carriedout in four phases. In the initialization phase, firefly search spacehas to be formed with the user defined bin details. The search spaceis in the form of network, each node in the network is linked witheach other node in the network. So that the firefly can move to anynode. Each node represents a bin with its packing constraints. Oncethe network has been formed, fireflies need to be initilized. Usinga pseudo-random number generator, the seed values for the fireflyintensities have been generated initially. These intensities will beupdated once the firefly started moving into the network. Updatingthe FFA control parameters are carried out using the Equations 4to 7. In order to reinforce further and to reduce computationaltime, both the single long training run (large number of items)and short multiple runs were performed to set the firefly algorithmcontrol parameters like α, ϵ, β etc., The next phase after initializingthe network is the training phase.

In the training phase, 500 iterations have been executed and ineach iteration 100 fireflies have been allowed to search the optimalsequence of bins to be packed. Initially, fireflies are allowed tosearch in the network at random and later based on the updated

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fitness values. The firefly movements can be in bidirectional andto any node without revisiting a node. The fireflies have to visitthe nodes, till the termination conditions reached. The terminationconditions considered in this research are the number of node visitshould be 1000 or fitness score should reach 1. As soon as theiterations are finished, the best bin packing sequence should bestored separately and the network is re-initialized for next iteration.During the evolutionary process of training, when evaluating thefitness for an individual firefly, for each and every bin selectionby the firefly, the penalty value also be calculated to obtain thefeasible solution. At each trial, a different instance (sequence ofbins) is produced for the same matrix of bins. The best instancesfrom the training phase are used to construct the updated networkand used to identify the optimal bin packing in the execution phase.

In the execution phase, the stored best sequences have beenused to formulate the network threshold values and the fireflies areallowed to search in the best network to obtain the better bin pack-ing sequence. Thus the algorithm provides a sequence with higherfitness function score, and the sequence have been consider as afeasible optimal bin packing pattern which performs packing of alarge number of bins by maximally utilizing the container volume.In the result phase, the developed module generated a sequence ofbins based on the firefly selection and packs in layer-by-layer pack-ing approach for top door containers and wall building approach forfront door containers. The report will be generated with the bindetails, its position inside the container along with the orientationand shipment details. The sample report is given in the Table I.

Similarly the bin packing sequence report has been generatedfor 1000 bins. This data given in the tabular format is also exploredin pictorial format for easy understanding of the user and the plan(top view) of layer 1 is shown in the Fig. 1. Each box in the Fig. 1represents the bin and the number inside those boxes represents thebin number as given by the user. Thus the user can understand theposition and orientation of each and every bin inside the containerin the Cartesian coordinate system. Similarly the developed modulegenerated the report for all the layers and for all the bins.

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6 Result and Discussion

In this study, an extensive analysis of the firefly methodology havebeen performed by applying it to a wider range of bin packinginstance generators and the observations are listed as follows.

1. By completing sensitivity analysis, it is found that the bestbin packing solution could be totally different for longer andshorter bin packing sequences.

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2. By increasing the number of iterations beyond 500 or increas-ing the number of fireflies in each iteration, the performanceimprovement reported from multiple runs on a given instanceis very small and negligible.

3. The effect of arraigning the bin matrix in user defined se-quence or in random fashion does not significantly affect theoverall results. But the performance after weight sorting andcategorizing the bins to form the network improves the result.

4. This study utilizes a fixed number of bin sequence, so insome cases, top most layer of the container may not filledcompletely with the bins and created unoccupied space. Sotraditional methodology of best fit algorithm had used to fillthe remaining unoccupied container space with the availableunpacked bins.

5. On the other hand, due to fixed instance, in some cases, con-tainer volume has been utilized fully by leaving some binsunpacked and the placement penalty function eliminates thatsequence. Thus the best sequence can also be eliminated.

6. The utilization of percentage of average bin fullness (over 500trials) achieved by each instance during the test phase for theexecution phase improves the result.

7. In this study, for completeness, the constraint violation hastested on the instance for every bin selection. Sometimes,it leads to redundant loop or consumes more computationaltime. Also the constraint checking can also be done at theend of all generation to reduce the computational time. Theexperimental results proved that the former methodology pro-duces better result compared to the later methodology of con-straint checking.

Table II gives the comparison of container volume utilization per-centage of various approaches such as best fit method, Heuristicgenetic Algorithm and Firefly algorithm for Bischoff and Ratcliff(1995) test instances. The data are the best value taken among fiveiterations.

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From Table 2, it is clear that, 22.84 % of the unpacked bins werepacked by the FFA into the container compared to other methods.Thus it is clear that the effectiveness of the FFA.

7 Conclusion

In this study, firefly framework has been used for identifying the op-timal bin packing sequence by considering the practical constraints.Experiments with 500 benchmark problem instances are carried outto verify the performance of the algorithms and found the resultsare satisfactory. The resulting instances are specialized and are

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much more effective with the tabular and graphical format whichcan be easily understood by the user. In order to reduce the man-ual interventions, no assumptions on the structure of solutions hadbeen made. The generic heuristics best fit have been used to rein-force the obtained result. Thus the container volume can be utilizedmaximally and thereby reducing the non-value added cost on theproduct using firefly algorithm.

Instead of having fixed number of instances in the sequence, itcan be dynamically varying instances to make the module inde-pendent of the user inputs. In this research, four different shapeshave been considered and due to stability constraint most of thespherical bins were eliminated from packing. So heuristics can bedeveloped to pack spherical bins. Positioning and number of pal-lets for easy loading and unloading have not optimized, instead, thepallet dimensions were added along with the bin dimensions.

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url: Special Issue ijpam.eu A Complete ...acadpubl.eu/jsi/2017-114-5/articles/2/31.pdf · used to generalize the behavior of genetic approach [3]. In most of the evolutionary and